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CHAPTER XII

TOPOLOGY AND TOPOLOGICAL ALGEBRA

As we said in the Introduction, we have sought to limit the material in this
chapter to the minimum that the reader will require. We shall not stay to
explore the refinements of general topology (filters, uniformities, separation
axioms); instead, we shall pass as soon as possible to the category of uni-
formizable spaces, which are the only ones we shall meet in later chapters.
Usually these spaces occur merely as " ambient spaces" in which it is con-
venient to operate, and we shall be mainly concernedóin conformity with the
spirit of this bookówith their separable metrizable subspaces. for this
reason we have included many criteria of metrizability and separability
(12.3.6,12.4.6,12.4.7,12.5.8,12.9.1,12.10.10,12.11.3, 12.14.6.2,12.15.7,12.15.9,
12.15.10). These, together with Baire's theorem and its consequences, are
the only results in the chapter whose proofs are not straightforward. We
have also included an account of various purely topological techniques which
were not needed in the first volume (partitions of unity (12.6), semi-continuous
functions (12.7)), and more than half the chapter is devoted to elementary
concepts of topological algebra (topological groups, spaces with operators,
topological vector spaces). All of this will be u^d constantly in the succeeding
chapters.

1. TOPOLOGICAL SPACES

A topology on a set E is a set C of subsets of E (in other words, a subset of
satisfying the following two conditions :

The union of any family (AA)A 6 L of sets belonging to D belongs to SO ;
(On)   The set E belongs to O, and the intersection of any two sets belonging
to O belongs to C.)                                   (p(x*x))i/2: 15.4, Problem 18